Difference between revisions of "Subgroup"
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− | A '''subgroup''' is a [[group]] contained in another. Specifically, let <math>H</math> and <math>G</math> be groups | + | A '''subgroup''' is a [[group]] contained in another. Specifically, let <math>H</math> and <math>G</math> be groups. We say that <math>H</math> is a subgroup of <math>G</math> if the [[element]]s of <math>H</math> are a [[subset]] of the [[set]] of elements of <math>G</math> and the group law on <math>H</math> agrees with group law on <math>G</math> where both are defined. We may denote this by <math>H \subseteq G</math> or <math>H \le G</math>. |
We say that <math>H</math> is a ''proper subgroup'' of <math>G</math> if <math>H \neq G</math>. | We say that <math>H</math> is a ''proper subgroup'' of <math>G</math> if <math>H \neq G</math>. | ||
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2&2&3&0&1 \ | 2&2&3&0&1 \ | ||
3&3&0&1&2 \end{array}</cmath> | 3&3&0&1&2 \end{array}</cmath> | ||
− | there are three subgroups : the group itself, <math>\{ 0 \}</math>, and the group <math>2 \mathbb{Z}/4\mathbb{Z}</math>, shown below. This last subgroup is [[isomorphic]] to the additive group <math>\mathbb{Z}/2\mathbb{Z}</math>. | + | there are three subgroups : the group itself, <math>\{ 0 \}</math>, and the group <math>2 \mathbb{Z}/4\mathbb{Z} = \{0, 2\}</math>, shown below. This last subgroup is [[isomorphic]] to the additive group <math>\mathbb{Z}/2\mathbb{Z}</math>. |
<cmath> \begin{array}{c|cc} & 0& 2 \\hline | <cmath> \begin{array}{c|cc} & 0& 2 \\hline | ||
0&0&2 \ 2&2&0 \end{array} </cmath> | 0&0&2 \ 2&2&0 \end{array} </cmath> | ||
− | Every group is the largest subgroup of itself. | + | Every group is the largest subgroup of itself. The set consisting of the [[identity]] element of a group is the smallest subgroup of that group. |
− | In a group <math>G</math>, the intersection of a family of subgroups of <math>G</math> is a subgroup of <math>G</math>. Thus for any collection <math>X</math> of elements of <math>G</math>, there exists a smallest subgroup containing these elements. This is called the subgroup generated by <math>X</math>. | + | In a group <math>G</math>, the [[intersection]] of a family of subgroups of <math>G</math> is a subgroup of <math>G</math>. Thus for any collection <math>X</math> of elements of <math>G</math>, there exists a smallest subgroup containing these elements. This is called the ''subgroup generated by'' <math>X</math>. |
− | In the additive group <math>\mathbb{Z}</math>, all subgroups are of the form <math>n \mathbb{Z}</math> | + | In the additive group <math>\mathbb{Z}</math>, all subgroups are of the form <math>n \mathbb{Z}</math> for some integer <math>n</math>. In particular, for <math>n=1</math> we have the integers themselves and for <math>n=0</math> we have <math>\{0\}</math>. |
== See Also == | == See Also == |
Revision as of 14:01, 20 February 2008
This article is a stub. Help us out by expanding it.
A subgroup is a group contained in another. Specifically, let and be groups. We say that is a subgroup of if the elements of are a subset of the set of elements of and the group law on agrees with group law on where both are defined. We may denote this by or .
We say that is a proper subgroup of if .
Examples
In the additive group , shown below, there are three subgroups : the group itself, , and the group , shown below. This last subgroup is isomorphic to the additive group .
Every group is the largest subgroup of itself. The set consisting of the identity element of a group is the smallest subgroup of that group.
In a group , the intersection of a family of subgroups of is a subgroup of . Thus for any collection of elements of , there exists a smallest subgroup containing these elements. This is called the subgroup generated by .
In the additive group , all subgroups are of the form for some integer . In particular, for we have the integers themselves and for we have .